To solve this expression, we need to follow the order of operations (PEMDAS/BODMAS).

First, let's simplify the subtraction inside the parentheses:
13 1/4 - 2 5/27 - 10 5/6

To subtract mixed numbers, we need to convert them into improper fractions:
13 1/4 = (13 * 4 + 1) / 4 = 53/4
2 5/27 = (2 * 27 + 5) / 27 = 59/27
10 5/6 = (10 * 6 + 5) / 6 = 65/6

Now, let's substitute these values back into the expression:
(53/4) - (59/27) - (65/6)

To subtract fractions, we need to find a common denominator. In this case, the least common multiple (LCM) of 4, 27, and 6 is 216.

(53/4) * (54/54) = 2862/216
(59/27) * (8/8) = 472/216
(65/6) * (36/36) = 2340/216

Now, let's substitute these values back into the expression:
(2862/216) - (472/216) - (2340/216)

To subtract fractions with the same denominator, we can simply subtract the numerators:
(2862 - 472 - 2340) / 216

Now, let's simplify the numerator:
2862 - 472 - 2340 = 50

Now, let's substitute this value back into the expression:
50/216

Next, let's simplify the addition outside the parentheses:
2 1/25 + 4 3/4

To add mixed numbers, we need to convert them into improper fractions:
2 1/25 = (2 * 25 + 1) / 25 = 51/25
4 3/4 = (4 * 4 + 3) / 4 = 19/4

Now, let's substitute these values back into the expression:
(51/25) + (19/4)

To add fractions, we need to find a common denominator. In this case, the least common multiple (LCM) of 25 and 4 is 100.

(51/25) * (4/4) = 204/100
(19/4) * (25/25) = 475/100

Now, let's substitute these values back into the expression:
(204/100) + (475/100)

To add fractions with the same denominator, we can simply add the numerators:
(204 + 475) / 100 = 679/100

Now, let's substitute this value back into the expression:
679/100

Finally, let's multiply the two results together:
(50/216) * (679/100)

To multiply fractions, we simply multiply the numerators and denominators:
(50 * 679) / (216 * 100) = 33950/21600

Now, let's simplify the fraction:
33950/21600 = 16975/10800

Therefore, the final result is 16975/10800.